Theorem nchoicelem7 6295

Description: Lemma for nchoice 6308. Calculate the cardinality of a special set generator. Theorem 6.7 of [Specker] p. 974. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
nchoicelem7	|- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Nc ( Spac ` M) = ( Nc ( Spac ` (2c ↑c M)) +o 1c))

Proof of Theorem nchoicelem7

Step	Hyp	Ref	Expression
1		nchoicelem6 6294	. . 3 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> ( Spac ` M) = ({M} u. ( Spac ` (2c ↑c M))))
2	1	nceqd 6110	. 2 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Nc ( Spac ` M) = Nc ({M} u. ( Spac ` (2c ↑c M))))
3		incom 3448	. . . . . 6 |- ({M} i^i ( Spac ` (2c ↑c M))) = (( Spac ` (2c ↑c M)) i^i {M})
4		nchoicelem5 6293	. . . . . . 7 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> -. M e. ( Spac ` (2c ↑c M)))
5		disjsn 3786	. . . . . . 7 |- ((( Spac ` (2c ↑c M)) i^i {M}) = (/) <-> -. M e. ( Spac ` (2c ↑c M)))
6	4, 5	sylibr 203	. . . . . 6 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> (( Spac ` (2c ↑c M)) i^i {M}) = (/))
7	3, 6	syl5eq 2397	. . . . 5 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> ({M} i^i ( Spac ` (2c ↑c M))) = (/))
8		snex 4111	. . . . . 6 |- {M} e. _V
9		fvex 5339	. . . . . 6 |- ( Spac ` (2c ↑c M)) e. _V
10	8, 9	ncdisjun 6136	. . . . 5 |- (({M} i^i ( Spac ` (2c ↑c M))) = (/) -> Nc ({M} u. ( Spac ` (2c ↑c M))) = ( Nc {M} +o Nc ( Spac ` (2c ↑c M))))
11	7, 10	syl 15	. . . 4 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Nc ({M} u. ( Spac ` (2c ↑c M))) = ( Nc {M} +o Nc ( Spac ` (2c ↑c M))))
12		df1c3g 6141	. . . . . 6 |- (M e. NC -> 1c = Nc {M})
13	12	addceq1d 4389	. . . . 5 |- (M e. NC -> (1c +o Nc ( Spac ` (2c ↑c M))) = ( Nc {M} +o Nc ( Spac ` (2c ↑c M))))
14	13	adantr 451	. . . 4 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> (1c +o Nc ( Spac ` (2c ↑c M))) = ( Nc {M} +o Nc ( Spac ` (2c ↑c M))))
15	11, 14	eqtr4d 2388	. . 3 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Nc ({M} u. ( Spac ` (2c ↑c M))) = (1c +o Nc ( Spac ` (2c ↑c M))))
16		addccom 4406	. . 3 |- (1c +o Nc ( Spac ` (2c ↑c M))) = ( Nc ( Spac ` (2c ↑c M)) +o 1c)
17	15, 16	syl6eq 2401	. 2 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Nc ({M} u. ( Spac ` (2c ↑c M))) = ( Nc ( Spac ` (2c ↑c M)) +o 1c))
18	2, 17	eqtrd 2385	1 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> Nc ( Spac ` M) = ( Nc ( Spac ` (2c ↑c M)) +o 1c))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   u. cun 3207   i^i cin 3208  (/)c0 3550  {csn 3737  1_cc1c 4134  0_cc0c 4374   +o cplc 4375  ` cfv 4781  (class class class)co 5525   NC cncs 6088   Nc cnc 6091  2_cc2c 6094   ↑_c cce 6096   Sp_ac cspac 6273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-lec 6099  df-ltc 6100  df-nc 6101  df-2c 6104  df-ce 6106  df-spac 6274

This theorem is referenced by:  nchoicelem9  6297  nchoicelem12  6300  nchoicelem17  6305